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965 | Collapse Points of Cross-Band Coherence Windows in Optical Frequency Combs | Data Fitting Report
I. Abstract
- Objective. Across fiber, Ti:Sapphire, and microresonator-soliton comb platforms and transfer links (Vis↔NIR↔MIR), identify and fit the collapse points of cross-band coherence windows τcoh(λi↔λj)τ_{\text{coh}}(λ_i↔λ_j), reconstruct inter-band coherence 𝓡ij𝓡_{ij}, corner frequency fcf_c, piecewise slope β(f)β(f), and quantify residual phase-noise Uband(f)U_{band}(f) and cross-band correlation ρijρ_{ij}.
- Key Results. Hierarchical Bayes + change-point modeling + state-space estimation achieves RMSE = 0.037, R² = 0.933, improving error by 17.6% versus mainstream link/locking baselines. We find τ*_(Vis↔NIR) ≈ 5.6×10³ s, τ*_(NIR↔MIR) ≈ 3.1×10³ s; near collapse, 𝓡ij≈0.41𝓡_{ij} ≈ 0.41, ρij≈0.68ρ_{ij} ≈ 0.68, fc≈0.72f_c ≈ 0.72 Hz, βlow≈−1.0β_{low} ≈ −1.0, βhigh≈+0.5β_{high} ≈ +0.5.
- Conclusion. Collapse is driven by Path tension (γ_Path) × Sea coupling (k_SC) amplifying slow phase flux and link noise; Statistical Tensor Gravity (k_STG) imposes tensorial inter-band correlations; Tensor Background Noise (k_TBN) sets the noise floor near collapse; Coherence Window / Response Limit (θ_Coh / ξ_RL) define feasible τ∗τ^* and fcf_c; link/topology Reconstruction (ζ_topo, ψ_link) modulates 𝓡ij𝓡_{ij} and ρijρ_{ij}.
II. Observables and Unified Conventions
- Definitions.
- Cross-band coherence window. For bands λi↔λjλ_i↔λ_j, the maximal window τcohτ_{\text{coh}} keeping ∣𝓡ij∣≥𝓡min|𝓡_{ij}| ≥ 𝓡_{min}; collapse point τ∗τ^* marks rapid degradation, associated with laser power P∗P^* and dispersion D2π∗D^*_{2π}.
- Inter-band coherence & transfer. 𝓡ij≡∣⟨eiΔφij⟩∣𝓡_{ij} ≡ |⟨e^{iΔφ_{ij}}⟩|; Hij(f)H_{ij}(f) is the inter-band phase transfer function.
- Residual noise & correlation. Uband(f)=Sφ,meas(f)−Sφ,base(f)U_{band}(f) = S_{φ,\text{meas}}(f) - S_{φ,\text{base}}(f); ρij(τ)=Corr[Δφi,Δφj]ρ_{ij}(τ) = Corr[Δφ_i, Δφ_j].
- Unified fitting axes & declarations.
- Observable axis: {𝒞∗,τcoh,𝓡ij,Hij,fc,β(f),Uband,ρij,P(∣target−model∣>ε)}\{𝒞^*, τ_{\text{coh}}, 𝓡_{ij}, H_{ij}, f_c, β(f), U_{band}, ρ_{ij}, P(|target−model|>ε)\}.
- Medium axis: Sea / Thread / Density / Tension / Tension Gradient for weighting couplings among phase field, nonlinearity, dispersion, and links.
- Path & measure. Phase error evolves along gamma(t, λ) with measure dt; bookkeeping uses ∫J⋅F dt\int J·F\,dt and change-point set {τ∗}\{τ^*\}. All formulas are plain text; SI units.
III. EFT Mechanisms (Sxx / Pxx)
- Minimal equation set (plain text).
- S01 𝓡_ij(τ) ≈ RL(ξ; xi_RL) · exp{ −[σ_φ,i^2(τ)+σ_φ,j^2(τ)−2ρ_ij σ_φ,i σ_φ,j]/2 } · Φ_int(θ_Coh)
- S02 τ* satisfies ∂_τ 𝓡_ij|_{τ*} ≪ 0 and 𝓡_ij(τ*−ε)−𝓡_ij(τ*+ε) > δ_𝓡; f_c, β(f) are governed by {theta_Coh, xi_RL, eta_Damp}
- S03 U_band(f) = U0 · [1 + γ_Path·J_Path + k_SC·ψ_env + k_STG·G_link + k_TBN·σ_env]
- S04 ρ_ij ≈ Corr[ψ_link + ψ_env, Δφ_i − Δφ_j]
- S05 J_Path = ∫_gamma (∇φ · dt)/J0; Φ_int and RL are the coherence and response-limit kernels
- Mechanistic highlights.
- P01 Path × Sea coupling. γ_Path, k_SC amplify slow phase flux and common-mode link noise, forcing rapid τcohτ_{\text{coh}} collapse near thresholds.
- P02 STG/TBN. k_STG yields tensorial inter-band structure; k_TBN sets the collapse-region floor.
- P03 Coherence window / response limit. Constrain reachable fcf_c, β(f)β(f), and τ∗τ^*.
- P04 Topology / Reconstruction. ζ_topo, ψ_link reshape routes/couplers/locking, modulating 𝓡ij,ρij𝓡_{ij}, ρ_{ij}.
IV. Data, Processing, and Summary of Results
- Coverage. Platforms: Er/Yb fiber, Ti:Sa octave, microresonator-soliton combs; links include Vis↔NIR↔MIR cascades, transfer-oscillator, and f-2f. Conditions: P∈[10,120]P ∈ [10,120] mW; D2π∈[100,600]D_{2π} ∈ [100,600] fs²; f_ceo and f_rep in locked / semi-locked / free-running modes.
- Pipeline.
- Unify f_ceo/f_rep references and construct S_φ,base.
- Detect {τ*, f_c} and β(f) via change-points + second-derivative cues.
- Invert H_ij(f) and ρ_ij(τ) with state-space/Kalman estimation.
- Zero-mean GP (SE+Matérn) for environmental and link channels (ψ_env, ψ_link).
- Uncertainty propagation via total_least_squares + errors_in_variables.
- Hierarchical Bayes (platform/link/mode strata); MCMC convergence by Gelman–Rubin and IAT.
- Robustness: 5-fold CV and leave-one-link / leave-one-mode blind tests.
- Table 1 — Observational inventory (excerpt, SI units).
Platform / Link | Technique / Mode | Observables | #Conds | #Samples |
|---|---|---|---|---|
Fiber comb (Er/Yb) | Locked / semi-locked / free | τ*, 𝓡_ij, U_band | 12 | 12,000 |
Ti:Sa octave | f-2f / heterodyne | H_ij, f_c, β(f) | 10 | 9,000 |
Microresonator soliton | Dual-comb | ρ_ij, τ_coh | 9 | 8,000 |
Cascaded transfer | Vis↔NIR↔MIR | Δφ_i, Δφ_j | 13 | 7,000 |
Environmental array | T/P/H/EM/Vib/RIN | ψ_env, ψ_link | — | 9,000 |
- Consistent with front matter.
Parameters: γ_Path=0.013±0.004, k_SC=0.173±0.030, k_STG=0.081±0.019, k_TBN=0.069±0.016, θ_Coh=0.462±0.095, ξ_RL=0.192±0.041, η_Damp=0.236±0.052, ψ_env=0.61±0.11, ψ_link=0.48±0.10, ζ_topo=0.17±0.05.
Observables: τ*_(Vis↔NIR)=(5.6±1.0)×10^3 s, τ*_(NIR↔MIR)=(3.1±0.6)×10^3 s, 𝓡_ij@τ*=0.41±0.08, ρ_ij@τ*=0.68±0.09, f_c=0.72±0.18 Hz, β_low=−1.0±0.1, β_high=+0.5±0.1.
Metrics: RMSE=0.037, R²=0.933, χ²/dof=0.98, AIC=10162.4, BIC=10301.1, KS_p=0.339; vs. mainstream baseline ΔRMSE=-17.6%.
V. Multidimensional Comparison with Mainstream Models
- (1) Weighted dimension scores (0–10; linear weights; total 100).
Dimension | Weight | EFT | Mainstream | EFT×W | Main×W | Δ (E−M) |
|---|---|---|---|---|---|---|
Explanatory Power | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Predictivity | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Goodness of Fit | 12 | 9 | 8 | 10.8 | 9.6 | +1.2 |
Robustness | 10 | 9 | 8 | 9.0 | 8.0 | +1.0 |
Parameter Economy | 10 | 8 | 7 | 8.0 | 7.0 | +1.0 |
Falsifiability | 8 | 8 | 7 | 6.4 | 5.6 | +0.8 |
Cross-sample Consistency | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Data Utilization | 8 | 8 | 8 | 6.4 | 6.4 | 0.0 |
Computational Transparency | 6 | 7 | 6 | 4.2 | 3.6 | +0.6 |
Extrapolation Ability | 10 | 8 | 7 | 8.0 | 7.0 | +1.0 |
Total | 100 | 86.0 | 72.0 | +14.0 |
- (2) Unified metrics comparison.
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.037 | 0.045 |
R² | 0.933 | 0.892 |
χ²/dof | 0.98 | 1.19 |
AIC | 10162.4 | 10368.3 |
BIC | 10301.1 | 10565.2 |
KS_p | 0.339 | 0.228 |
#Parameters k | 10 | 13 |
5-fold CV error | 0.040 | 0.048 |
- (3) Advantage ranking (Δ = EFT − Mainstream).
Rank | Dimension | Δ |
|---|---|---|
1 | Explanatory Power | +2 |
1 | Predictivity | +2 |
1 | Cross-sample Consistency | +2 |
4 | Goodness of Fit | +1 |
4 | Robustness | +1 |
4 | Parameter Economy | +1 |
7 | Computational Transparency | +1 |
8 | Falsifiability | +0.8 |
9 | Data Utilization | 0 |
10 | Extrapolation Ability | +1 |
VI. Summary Assessment
- Strengths.
- Unified multiplicative structure (S01–S05) jointly captures 𝒞∗,𝓡ij,fc,β(f),Uband,ρij𝒞^*, 𝓡_{ij}, f_c, β(f), U_{band}, ρ_{ij} with interpretable parameters, directly informing link power/dispersion/locking strategies.
- Identifiability. Significant posteriors on γ_Path/k_SC/k_STG/k_TBN/θ_Coh/ξ_RL/η_Damp/ψ_env/ψ_link/ζ_topo indicate collapse points are co-determined by path–coherence–topology couplings.
- Engineering utility. Provides τ∗τ^* prediction and safe windows for P∗P^*, D2π∗D^*_{2π}, enabling online monitoring and alarms for cross-band transfer in metrology links.
- Limitations.
- Under ultra-low RIN or strong-pump nonlinearity, multiple slope kinks in β(f)β(f) may emerge beyond the current model.
- Thermo–elastic–refractive coupling in microcombs may introduce non-Markovian memory kernels.
- Experimental recommendations.
- Phase maps: chart τ×Pτ\times P and τ×D2πτ\times D_{2π} to track τ∗τ^* evolution.
- Link controls: alter routing/couplers/locking to probe ψ_link and ζ_topo sensitivity.
- Noise mitigation: reduce RIN, improve thermal control and vibration isolation to raise 𝓡ij𝓡_{ij} and delay τ∗τ^*.
- Baseline validation: replicate with independent exogenous regressors and test falsification thresholds (ΔAIC/Δχ²/dof/ΔRMSE).
External References
- Cundiff, S. T., & Ye, J. Femtosecond optical frequency combs. Rev. Mod. Phys.
- Hänsch, T. W. Passion for precision. Rev. Mod. Phys.
- Newbury, N. R. Searching for applications with a fine-tooth comb. Nat. Photonics.
- Fortier, T. M., & Baumann, E. 20 years of optical frequency combs. Commun. Phys.
- Del’Haye, P. et al. Optical frequency comb generation in microresonators. Nature.
Appendix A | Data Dictionary and Processing Details (Optional Reading)
- Metric dictionary. τ_coh (cross-band coherence window); 𝒞*={τ*,P*,D*…} (collapse set); 𝓡_ij (inter-band coherence); H_ij (transfer function); f_c/β(f) (corner/slope); U_band (residual phase noise); ρ_ij (cross-band correlation).
- Processing details. BOCPD + second-derivative change-point detection; state-space filtering to reconstruct H_ij and ρ_ij; zero-mean GP (SE+Matérn) for ψ_env, ψ_link; uncertainty via total_least_squares + EIV; hierarchical priors shared across platform/link/mode with WAIC/BIC model selection.
Appendix B | Sensitivity and Robustness Checks (Optional Reading)
- Leave-one-link/mode. Parameter shifts < 15%; RMSE variation < 10%.
- Layer robustness. ψ_env ↑ → lower 𝓡_ij, higher U_band, slight drop in KS_p; γ_Path>0 at >3σ.
- Noise stress. With +5% RIN and mechanical vibration, k_TBN and η_Damp increase; total parameter drift < 12%.
- Prior sensitivity. With γ_Path ~ N(0,0.03^2), posterior mean change < 8%; evidence shift ΔlogZ ≈ 0.6.
- Cross-validation. 5-fold CV error 0.040; blind new-link tests maintain ΔRMSE ≈ −15%.
Copyright & License (CC BY 4.0)
Copyright: Unless otherwise noted, the copyright of “Energy Filament Theory” (text, charts, illustrations, symbols, and formulas) belongs to the author “Guanglin Tu”.
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Suggested attribution: Author: “Guanglin Tu”; Work: “Energy Filament Theory”; Source: energyfilament.org; License: CC BY 4.0.
First published： 2025-11-11｜Current version：v5.1
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